// WARNING: Do not change or delete this file! If you do, Perseus might become
// unable to parse the current data format, which will break clients.
// If you need to add more regression tests, add a new file to this directory.
export default {
    question: {
        content:
            "Consider this matrix transformation:\n\n$\\left[\\begin{array}{c}\n0 & 2 & 1 & -3\n\\\\\\\\ \n-1 & 2 & -3 & 0\n\\\\\\\\\n2 & -2 & 0 &1\n\\\\\\\\\n1 & 1 & -1 & -1\n\\end{array}\\right]$\n\n**What is the image of $\\left[\\begin{array}{c}\n-5\n\\\\\\\\\n1\n\\\\\\\\\n3\n\\\\\\\\\n-2\n\\end{array}\\right]$ under this transformation?**\n\n[[☃ matrix 1]]",
        images: {},
        widgets: {
            "matrix 1": {
                type: "matrix",
                alignment: "default",
                static: false,
                graded: true,
                options: {
                    static: false,
                    matrixBoardSize: [4, 1],
                    answers: [["11"], [-2], [-14], [-5]],
                    prefix: "",
                    suffix: "",
                    cursorPosition: [0, 0],
                },
                version: {
                    major: 0,
                    minor: 0,
                },
            },
        },
    },
    answerArea: {
        calculator: true,
        chi2Table: false,
        periodicTable: false,
        tTable: false,
        zTable: false,
    },
    itemDataVersion: {
        major: 0,
        minor: 1,
    },
    hints: [
        {
            replace: false,
            content:
                "In general terms, suppose we have\n\n- an $n$-dimensional square matrix whose columns are $\\vec{v_1},\\vec{v_2},...,\\,\\vec{v_n}$ and\n- an $n$-dimensional vector $\\vec x=\\left[\\begin{array}{c}x_1\\\\\\\\x_2\\\\\\\\...\\\\\\\\x_n\\end{array}\\right]$,\n\nthen this is the image of the vector under the matrix transformations:\n\n$x_1\\cdot\\vec{v_1}+x_2\\cdot\\vec{v_2}+...+x_n\\cdot\\vec{v_n}$\n\n[[☃ explanation 1]]",
            images: {},
            widgets: {
                "explanation 1": {
                    type: "explanation",
                    alignment: "default",
                    static: false,
                    graded: true,
                    options: {
                        static: false,
                        showPrompt: "Why is this true?",
                        hidePrompt: "Got it, thanks!",
                        explanation:
                            "We learned that for 2-dimensional matrices, the first column is the image of the unit vector $\\left[\\begin{array}{c}1\\\\\\\\0\\end{array}\\right]$ and the second column is the image of the unit vector $\\left[\\begin{array}{c}0\\\\\\\\1\\end{array}\\right]$.\n\nThis can be generalized to higher dimensions, only now we have $n$ unit vectors which are full of zeros except for their $i^{\\text{th}}$ entry. For example, these are the four unit vectors in 4D:\n\n$\\left[\\begin{array}{c}1\\\\\\\\0\\\\\\\\0\\\\\\\\0\\end{array}\\right]\n\\left[\\begin{array}{c}0\\\\\\\\1\\\\\\\\0\\\\\\\\0\\end{array}\\right]\n\\left[\\begin{array}{c}0\\\\\\\\0\\\\\\\\1\\\\\\\\0\\end{array}\\right]\n\\left[\\begin{array}{c}0\\\\\\\\0\\\\\\\\0\\\\\\\\1\\end{array}\\right]$\n\nEach column of a 4D matrix tells us where it maps each unit vector, and we can write the general 4D vector $\\left[\\begin{array}{c}x_1\\\\\\\\x_2\\\\\\\\x_3\\\\\\\\x_4\\end{array}\\right]$ as:\n\n$x_1\\cdot\\left[\\begin{array}{c}1\\\\\\\\0\\\\\\\\0\\\\\\\\0\\end{array}\\right]\n+x_2\\cdot\\left[\\begin{array}{c}0\\\\\\\\1\\\\\\\\0\\\\\\\\0\\end{array}\\right]\n+x_3\\cdot\\left[\\begin{array}{c}0\\\\\\\\0\\\\\\\\1\\\\\\\\0\\end{array}\\right]\n+x_4\\cdot\\left[\\begin{array}{c}0\\\\\\\\0\\\\\\\\0\\\\\\\\1\\end{array}\\right]$",
                        widgets: {},
                    },
                    version: {
                        major: 0,
                        minor: 0,
                    },
                },
            },
        },
        {
            replace: false,
            content:
                "$\\begin{align}\n&\\phantom{=}\\left[\\begin{array}{c}\n\\tealE{0} & \\redE{2} & \\purpleE{1} & \\goldE{-3}\n\\\\\\\\ \n\\tealE{-1} & \\redE{2} & \\purpleE{-3} & \\goldE{0}\n\\\\\\\\\n\\tealE{2} & \\redE{-2} & \\purpleE{0} &\\goldE{1}\n\\\\\\\\\n\\tealE{1} & \\redE{1} & \\purpleE{-1} & \\goldE{-1}\n\\end{array}\\right]\n\\left(\\left[\\begin{array}{c}\n-5\n\\\\\\\\\n1\n\\\\\\\\\n3\n\\\\\\\\\n-2\n\\end{array}\\right]\\right)\n\\\\\\\\\n&=-5\\cdot \\left[\\begin{array}{c}\n\\tealE{0}\n\\\\\\\\ \n\\tealE{-1}\n\\\\\\\\\n\\tealE{2}\n\\\\\\\\\n\\tealE{1}\n\\end{array}\\right]+1\\cdot\\left[\\begin{array}{c}\n\\redE{2}\n\\\\\\\\ \n\\redE{2}\n\\\\\\\\\n\\redE{-2}\n\\\\\\\\\n\\redE{1}\n\\end{array}\\right]+3\\cdot \\left[\\begin{array}{c}\n\\purpleE{1}\n\\\\\\\\ \n\\purpleE{-3}\n\\\\\\\\\n\\purpleE{0}\n\\\\\\\\\n\\purpleE{-1}\n\\end{array}\\right]+(-2)\\cdot\\left[\\begin{array}{c}\n\\goldE{-3}\n\\\\\\\\ \n\\goldE{0}\n\\\\\\\\\n\\goldE{1}\n\\\\\\\\\n\\goldE{-1}\n\\end{array}\\right]\n\\\\\\\\\n&=\\left[\\begin{array}{c}\n0\n\\\\\\\\ \n5\n\\\\\\\\\n-10\n\\\\\\\\\n-5\n\\end{array}\\right]+\\left[\\begin{array}{c}\n2\n\\\\\\\\ \n2\n\\\\\\\\\n-2\n\\\\\\\\\n1\n\\end{array}\\right]+\\left[\\begin{array}{c}\n3\n\\\\\\\\ \n-9\n\\\\\\\\\n0\n\\\\\\\\\n-3\n\\end{array}\\right]+\\left[\\begin{array}{c}\n6\n\\\\\\\\ \n0\n\\\\\\\\\n-2\n\\\\\\\\\n2\n\\end{array}\\right]\n\\\\\\\\\n&=\\left[\\begin{array}{c}\n11\n\\\\\\\\\n-2\n\\\\\\\\\n-14\n\\\\\\\\\n-5\n\\end{array}\\right]\n\\end{align}$",
            images: {},
            widgets: {},
        },
        {
            replace: false,
            content:
                "This is the image we obtain when we perform the transformation $\\left[\\begin{array}{c}\n0 & 2 & 1 & -3\n\\\\\\\\ \n-1 & 2 & -3 & 0\n\\\\\\\\\n2 & -2 & 0 &1\n\\\\\\\\\n1 & 1 & -1 & -1\n\\end{array}\\right]$ on the pre-image $\\left[\\begin{array}{c}\n-5\n\\\\\\\\\n1\n\\\\\\\\\n3\n\\\\\\\\\n-2\n\\end{array}\\right]$:\n\n$\\left[\\begin{array}{c}\n11\n\\\\\\\\\n-2\n\\\\\\\\\n-14\n\\\\\\\\\n-5\n\\end{array}\\right]$",
            images: {},
            widgets: {},
        },
    ],
};
